# Definition:Special Linear Group

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## Definition

Let $R$ be a commutative ring with unity whose zero is $0$ and unity is $1$.

The **special linear group of order $n$ on $R$** is the set of square matrices of order $n$ whose determinant is $1$.

It is a group under (conventional) matrix multiplication.

It is denoted $\SL {n, R}$, or $\SL n$ if the ring is implicit.

The ring itself is usually a standard number field, but can be *any* commutative ring with unity.

## Also denoted as

Some authors prefer $\map {\mathrm {SL}_n } R$ and $\operatorname{SL}_n$ for the **special linear group over $\SL {n, R}$.**

## Also see

- Results about
**the special linear group**can be found**here**.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 36$. Subgroups - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**general linear group** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**general linear group**